All nuclear weapons states (the US, USSR, UK, France, China, India, Pakistan, North Korea) have tested nuclear weapons except perhaps two (Israel and South Africa, the latter only a former weapons state now, may have had some kind of clandestine tests). The US and USSR tested hundreds of nuclear weapons during the Cold War.

You can break the motivations for nuclear weapons testing down into roughly three categories:

• weapons design tests: "does this approach to building a nuclear weapon actually work? how much output do you get from it?" — the first such test was the Trinity test in 1945, but this made up a lot of the nuclear testing by the USA and USSR.

• weapons effects tests: "what happens to other stuff if you nuke it?" — this made up a significant fraction of the testing as well (you might think it is "obvious" what happens if you nuke something, but it's not, if you care about the details) — the first such test series was Operation Crossroads, which is responsible for the image you have linked to (the question being asked was, "what happens to a fleet if you nuke it? what happens if you set the nuke off under water?" Answer: boats near the blast get destroyed but those just a little way out do OK, except for the huge amounts of radioactive water your nuke creates.)

• physics experiments: nuclear weapons work in energy regimes not normally accessible to human beings (millions of degrees of heat, for example, which render heavy metals as if they were gasses). If you really want to know what's going on "under the hood" at a deep level, there are sometimes experiments you have to do that only a nuke can do for you. A lot of the later US tests were basically "model refinement" tests where they tested parts of their understanding of how the physics worked.

• and a few other miscellaneous categories (e.g., safety tests — if we try to simulate conditions under which this nuke would be accidentally detonated, what happens?; plowshares tests — can we dig holes with nukes?; military exercises — do troops lose their minds when they see a nuke go off?; reliability tests — if we take a random warhead from the stockpile and set it off, does it behave as expected?)

In practice any given test could be a bunch of motivations simultaneously (you could test a new weapon design and effects on things and get basic science data at the same test, though in some cases matching these up was tricky if you weren't 100% sure what the weapon itself would do), and could also be done for "cultural" reasons beyond the science (e.g., Crossroads was also very much about making a spectacle out of the test — observers from many nations, including the USSR, and journalists were all invited). Some tests were derided as being exclusively "political" (e.g., a form of sabre-rattling) though in general they had some scientific "output" as well.

The US tested most of its weapons in Nevada at the Nevada Test Site, and most of them underground (post-1963, all US and Soviet tests were underground). Most of the photos one sees are the atmospheric (aboveground) tests, and many of them are in the Pacific as /u/b1luepenguin has outlined. Typically states tested "smaller" bombs at home and bigger ones in these more isolated sites.

These atmospheric tests, it is worth noting, often created a lot of radioactive debris that would move with the wind, and is now thought to have contributed to many thousands of fatal cancers.

You can actually go scuba diving to see most of those wrecks if you were ever so inclined and not afraid of potential exposure to radiation (supposedly its safe enough). The photo itself is from the 1946 Baker shot during Operation Crossroads (and is nicely colorized, I've only ever seen black and white versions). The ships pictured are a mix of decommissioned and captured ships from the Second World War and were sunk as a result of the nuclear testing during Operation Crossroads in the center of Bikini Atoll.

These tests were the first to take place after the end of the war and were to establish the efficacy of the bomb; especially its effects on a fleet of ships (would they be sunk, melted, or otherwise rendered hot from radiation-- and if so, could they be quickly decontaminated). The tests were sold to the Bikini Islanders by Commodore Ben Wyatt as part of their Christian moral obligation, that by surrendering their island temporarily, they would be making a sacrifice "for the good of all mankind." Whether or not the islanders would have actually been given a choice had they declined... well much like French testing later on Moruroa, the decision had already effectively been made whether the local people or government approved.

Ultimately Bikini Atoll was considered part of the Pacific Proving Grounds; a faraway region that no one in the US had really heard of where atomic weapons could safely be tested out of sight out of mind (the Soviets will have a similar policy with regards to testing in Siberia). Between 1946 to 1962; 105 nuclear weapons will be detonated in the proving grounds-- which are still contaminated to this day.

The tests were controversial even at the time; Bikini Atoll is part of the Marshall Islands, which were part of the Trust Territories of the Pacific, island groups being government by the US after the conclusion of World War Two. Theoretically, as a Trust Territory, the US was responsible for the economic and political development of the islands, their people, and their resources, until such time as they had achieved independence. In reality, dependency not development, was the ultimate goal of military officials who feared the spread of communism into the South Pacific and saw the islands as key strategic areas-- in the case of the Marshall islands, for nuclear bomb testing, and later on missile testing (and the Marshall islands are still used as such to this day).

The US was of course not alone in testing nuclear weapons in the Pacific. The British carried out testing in remote parts of Australia, areas that were supposedly scarcely populated (places likely to only be entered or inhabited by Aborigines, which only counted as people for some British officials.) Concern for public well-being was not high on the list for nuclear testing anywhere in the Pacific and the British did not always take into account the weather when exploding bombs, allowing radiated dust to spread into 'proper White Australian settlements'. People with no say and no knowledge of what was happening were exposed to extremely high levels of radiation and of course the same was true for American testing in the Marshall Islands. Not only were indigenous populations exposed (and continue to suffer the consequences of such expose) but US, British, and French military personnel were exposed during the experiments (though in most cases the governments deny exposing personnel to unsafe levels).

In total; with the final conclusion of nuclear weapons testing in the Pacific in 1996, 339 nuclear tests had been conducted ranging from the 15 megaton hydrogen bomb Castle Bravo, to small bombs like Baker Shot's 26 kilotons.

105 tests were conducted by the US in the Pacific (only a fraction of the total number of tests the US conducted, but representing the majority of the total megatonnage detonated-- smaller bombs were tested on the continental US), 189 French tests, and 45 British tests.

So, this is the Pacific context anyway. Continental testing, in the case of the US and Russia are not a topic I know quite as well.

Some reading if you feel so moved :

• Barker, Holly. Bravo for the Marshallese: Regaining Control in a Post-Nuclear/Colonial World. Belmont, California: Wadsworth Cengage Learning, 2013.

• Dibblin, Jane. Day of Two Suns: US Nuclear Testing and the Pacific Islanders. New York, New York: New Amsterdam, 1990.

• Maclellan, Nic. After Moruroa: France in the South Pacific. New York, New York: Ocean Press, 1998.

• Niedenthal, Jack. For the Good of Mankind: A History of the People of Bikini and Their Islands. Majuro, Marshall Islands: Micronitor Publishing, 2001.

• Weisgall, Jonathan. Operation Crossroads: The Atomic Tests at Bikini Atoll. Annapolis: Naval Institute Press, 1994.